Hello Taylor. I would like to add to Carlo's answer. These words, as many others, will share certain semantical coincidence, but their whole scope will highly depend on the region. For example the word "villa" in Chile may mean a upper-class neighbourhood, but in Argentina, a "villa" is usually referred to a poor area with a very negative connotation. Giving birth to the word "villero", someone from a "villa", which in Argentina may be used to speak of a poor, uneducated, theaf... But it would not be used to refer to a person who may also live in a villa but is an honest person.
Having said that, from the strict point of view of the original meaning of the words you are asking about, "vencindad" and "vencindario" come from the word "vecino", meaning "neighbour" in Spanish, which, in turn, comes from "vicinus", Latin for "around" or "close". Therefore, "una vecindad" and "un vecindario" are places where people is close to each other, therefore both words may be used to decribe an area where houses are close to each other. It would not be used to describe a community, that even being a community, houses are far apart. Though the people living there, they will call themselves "vecino". Barrio, comes from Arabic "barri", meaning "exterior". The word was used to describe the houses or any type of comlex of buildings that was immediately outsie the city walls. So, "barrio" had the idea of describing places where people could, live, go to chuch or work, but that was not inside the city walls but outside. Therefore, a "barrio" will normally refer to districts of a city that are not the centre one. Even if there is an area of the centre where people live, that area would not referred to as a "barrio" just because it is in the centre of the city, but it could perfectly be called a "vecindario" or "vecindad". So, now you can see that a neibourhood in Spanish could be a barrio or a vecindario at the same time... Or only one of them... Or even neither of them.